Finite Group Databasebeta

← #469×D173 #36363155

Subgroups — #469×D173 #36363155

Order 22144.

596 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D8×D173C211yesmaximal, characteristic, complemented
2C2×M16×D173C211yesmaximal, characteristic, complemented
3#121×D173C212yesmaximal, complemented
4#26011745C211yesmaximal, characteristic, complemented
5#26011744C211yesmaximal, characteristic, complemented
6#26011743C212yesmaximal, complemented
7#26011742C212yesmaximal, complemented
8#26011741C211yesmaximal, characteristic, complemented
9#106×D173C211yesmaximal, characteristic, complemented
10#102×D173C212yesmaximal, complemented
11#469×C173C211yesmaximal, characteristic, complemented
12C2×D4×D173—22no—
13C2×D4×D173C2212yes—
14C22×C4×D173C2211yescharacteristic
15C2×D1384C2211yescharacteristic, complemented
16C2×#14583811C2212yescomplemented
17D8×D173—24no—
18C2×#14583810C2211yescharacteristic, complemented
19C2×#14583809C2211yescharacteristic, complemented
20C2×#14583809C2211yescharacteristic, complemented
21M16×D173—24no—
22C2×C8×D173C2211yescharacteristic
23C2×C8×D173C2211yescharacteristic
24#19149654C2212yescomplemented
25#19149653C2212yescomplemented
26#19149652—22no—
27#19149651—22no—
28#19149650C2212yescomplemented
29C4⋊C4×D173C2212yes—
30C22⋊C4×D173—22no—
31#19149611C2211yescharacteristic, complemented
32#19149610C2212yescomplemented
33#19149601C2212yescomplemented
34#19149600C2212yescomplemented
35#19149599C2212yescomplemented
36C2×D8×C173C2211yescharacteristic, complemented
37C2×M16×C173C2211yescharacteristic, complemented
38#121×C173C2212yescomplemented
39#106×C173C2211yescharacteristic, complemented
40#102×C173C2212yescomplemented
41C23×D173—22no—
42C23×D173D411yescharacteristic
43C2×D692—22no—
44C2×D692C2312yes—
45C2×C22⋊D173—22no—
46C2×C22⋊D173—22no—
47C2×C22⋊D173—22no—
48D4×D173—24no—
49D4×D173—24no—
50D4×D173—48no—
51C22×Dic173D411yescharacteristic
52C2×C4×D173—22no—
53C2×C4×D173—22no—
54C2×C4×D173—22no—
55C2×C4×D173D411yescharacteristic
56C2×C4×D173D411yescharacteristic
57C2×C4×D173C2311yescharacteristic
58C2×C4×D173C2311yescharacteristic
59D1384—22no—
60#14583811—24no—
61#14583810—22no—
62#14583809—22no—
63#14583809—22no—
64C2×#13063436C2311yescharacteristic
65C2×#13063436C2311yescharacteristic
66C8×D173—22no—
67C8×D173D414yescomplemented
68C4⋊Dic173C2312yes—
69#14583805—22no—
70C22⋊Dic173—22no—
71#14583803—22no—
72#14583803—22no—
73C2×D4×C173—22no—
74C2×D4×C173C2312yes—
75C22×C4×C173C2311yescharacteristic
76D8×C173—22no—
77M16×C173—22no—
78C2×C8×C173C2311yescharacteristic
79C2×C8×C173C2311yescharacteristic
80C4⋊C4×C173C2312yes—
81C22⋊C4×C173—22no—
82C22×D173—22no—
83C22×D173—24no—
84C22×D173—22no—
85C22×D173—44no—
86C22×D173—44no—
87C22×D173—44no—
88C22×D173C2×D411yescharacteristic
89C22×D173C2×D411yescharacteristic
90D692—22no—
91D692—22no—
92D692—44no—
93C22⋊D173—44no—
94C22⋊D173—44no—
95C22⋊D173—44no—
96C2×Dic173—22no—
97C2×Dic173—22no—
98C2×Dic173C2×D411yescharacteristic
99C2×Dic173C2×D411yescharacteristic
100C4×D173—22no—
101C4×D173—21no—
102C4×D173—21no—
103C4×D173—22no—
104C4×D173—44no—
105C4×D173C2×D412yes—
106C4×D173#4112yes—
107#13063436—21no—
108#13063436C2×D412yescomplemented
109C23×C173—22no—
110C23×C173C2×D411yescharacteristic
111D4×C173—22no—
112D4×C173—22no—
113D4×C173—44no—
114C2×C4×C173—21no—
115C2×C4×C173—21no—
116C2×C4×C173—22no—
117C2×C4×C173C2×D411yescharacteristic
118C2×C4×C173C2411yescharacteristic
119C8×C173—21no—
120C8×C173C2×D412yescomplemented
121C22×C173—21no—
122C22×C173—22no—
123C22×C173—42no—
124C22×C173—42no—
125C22×C173—42no—
126C22×C173C22×D411yescharacteristic
127C2×D173—21no—
128C2×D173—22no—
129C2×D173—44no—
130C2×D173—42no—
131C2×D173—42no—
132C2×D173—42no—
133C2×D173—88no—
134C2×D173#12112yes—
135C2×D173AGL(1,ℤ/8)14yes—
136C4×C173—21no—
137C4×C173—42no—
138C4×C173C22×D411yescharacteristic
139C4×C173C2×#4111yescharacteristic
140Dic173—21no—
141Dic173—42no—
142Dic173#12112yes—
143C2×C173—41no—
144C2×C173—82no—
145C2×C173C2×#12111yescharacteristic
146C2×C173C2×AGL(1,ℤ/8)12yes—
147D173—41no—
148D173—82no—
149D173#46914yesdirect product factor
150C173C2×#46911yescharacteristic, complemented, Sylow
151C2×#469—1731nomaximal, Sylow
152C22×D8—1731no—
153C22×M16—1731no—
154C2×#121—1732no—
155#469—1731no—
156#469—1731no—
157#469—1732no—
158#469—1731no—
159#469—1732no—
160#469D17311yescharacteristic, direct product factor
161C2×#106—1731no—
162C2×#102—1732no—
163C22×D4—3462no—
164C22×D4—1732no—
165C23×C4—1731no—
166C2×D8—3464no—
167C2×D8—1732no—
168C2×D8—1731no—
169C2×D8C2×D17311yescharacteristic, complemented
170C2×M16—3464no—
171C2×M16—1731no—
172C2×M16—1731no—
173C2×M16—1731no—
174C2×M16C2×D17311yescharacteristic, complemented
175C22×C8—1731no—
176C22×C8—1731no—
177#121—3462no—
178#121—3462no—
179#121—1732no—
180#121—1732no—
181#121—1732no—
182#121C2×D17312yescomplemented
183C2×C4⋊C4—1732no—
184C2×C22⋊C4—3462no—
185#106—1731no—
186#106—1732no—
187#106C2×D17311yescharacteristic, complemented
188#102—1732no—
189#102—1732no—
190#102—1732no—
191#102C2×D17312yescomplemented
192C24—3462no—
193C24—1731no—
194C2×D4—22no—
195C2×D4—3462no—
196C2×D4—3462no—
197C2×D4—3462no—
198C2×D4—3464no—
199C2×D4—3464no—
200C2×D4—3462no—
201C2×D4—1732no—
202C2×D4—6928no—
203C2×D4C22×D17312yes—
204C22×C4—3462no—
205C22×C4—3462no—
206C22×C4—3462no—
207C22×C4—1731no—
208C22×C4—1731no—
209C22×C4—1731no—
210C22×C4—1731no—
211C22×C4—1731no—
212C22×C4C22×D17311yescharacteristic
213D8—22no—
214D8—3464no—
215D8—3462no—
216M16—22no—
217M16—3462no—
218M16—3462no—
219M16—3462no—
220C2×C8—3462no—
221C2×C8—1731no—
222C2×C8—1731no—
223C2×C8—1734no—
224C2×C8C22×D17311yescharacteristic
225C2×C8C22×D17311yescharacteristic
226C4⋊C4—3462no—
227C4⋊C4—1732no—
228C4⋊C4C22×D17312yes—
229C22⋊C4—22no—
230C22⋊C4—3462no—
231C22⋊C4—3462no—
232C22⋊C4—3462no—
233C23—22no—
234C23—3462no—
235C23—3464no—
236C23—3462no—
237C23—1731no—
238C23—1731no—
239C23—6924no—
240C23—6924no—
241C23—6924no—
242C23D4×D17311yescharacteristic
243D4—22no—
244D4—22no—
245D4—44no—
246D4—3462no—
247D4—3462no—
248D4—6924no—
249D4—6924no—
250D4—6924no—
251D4—6924no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—3462no—
256C2×C4—3462no—
257C2×C4—3462no—
258C2×C4—3461no—
259C2×C4—3461no—
260C2×C4—3462no—
261C2×C4—1731no—
262C2×C4—1731no—
263C2×C4—1732no—
264C2×C4—1732no—
265C2×C4—6924no—
266C2×C4D4×D17311yescharacteristic
267C2×C4C23×D17311yescharacteristic
268C8—21no—
269C8—3461no—
270C8—1732no—
271C8D4×D17312yescomplemented
272C22—21no—
273C22—22no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—3461no—
278C22—3462no—
279C22—13848no—
280C22—1732no—
281C22—1734no—
282C22—6924no—
283C22—6922no—
284C22—6922no—
285C22—6922no—
286C22C2×D4×D17311yescharacteristic
287C4—21no—
288C4—42no—
289C4—3461no—
290C4—1732no—
291C4—6922no—
292C4C2×D4×D17311yescharacteristic
293C4#41×D17311yescharacteristic
294C2—41no—
295C2—82no—
296C2—13842no—
297C2—1734no—
298C2—6921no—
299C2#121×D17311yescharacteristic
300C2AGL(1,ℤ/8)×D17312yes—
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#469×C173C2
socleC22×C173C22×D4
hypercenter#469D173
hypocenterC173C2×#469
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×D8×D173C2semidirect
2. C2×M16×D173C2semidirect
3. #121×D173C2semidirect
4. #26011745C2semidirect
5. #26011744C2semidirect
6. #26011743C2semidirect
7. #26011742C2semidirect
8. #26011741C2semidirect
9. #106×D173C2semidirect
10. #102×D173C2semidirect
11. #469×C173C2semidirect
15. C2×D1384C22semidirect
16. C2×#14583811C22semidirect
18. C2×#14583810C22semidirect
19. C2×#14583809C22semidirect
20. C2×#14583809C22semidirect
24. #19149654C22semidirect
25. #19149653C22semidirect
28. #19149650C22semidirect
31. #19149611C22semidirect
32. #19149610C22semidirect
33. #19149601C22semidirect
34. #19149600C22semidirect
35. #19149599C22semidirect
36. C2×D8×C173C22semidirect
37. C2×M16×C173C22semidirect
38. #121×C173C22semidirect
39. #106×C173C22semidirect
40. #102×C173C22semidirect
67. C8×D173D4semidirect
108. #13063436C2×D4semidirect
120. C8×C173C2×D4semidirect
149. D173#469direct (N × H)
150. C173C2×#469semidirect
160. #469D173direct (N × H)
169. C2×D8C2×D173semidirect
174. C2×M16C2×D173semidirect
182. #121C2×D173semidirect
187. #106C2×D173semidirect
191. #102C2×D173semidirect
271. C8D4×D173semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×D8×D1732
C2×M16×D1732
#121×D1732
#260117452
#260117442
#260117432
#260117422
#260117412
#106×D1732
#102×D1732
#469×C1732
C2×#469173